1/* origin: FreeBSD /usr/src/lib/msun/src/e_rem_pio2.c */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunSoft, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 *
12 * Optimized by Bruce D. Evans.
13 */
14/* __rem_pio2(x,y)
15 *
16 * return the remainder of x rem pi/2 in y[0]+y[1]
17 * use __rem_pio2_large() for large x
18 */
19
20#include "libm.h"
21
22#if FLT_EVAL_METHOD==0 || FLT_EVAL_METHOD==1
23#define EPS DBL_EPSILON
24#elif FLT_EVAL_METHOD==2
25#define EPS LDBL_EPSILON
26#endif
27
28/*
29 * invpio2: 53 bits of 2/pi
30 * pio2_1: first 33 bit of pi/2
31 * pio2_1t: pi/2 - pio2_1
32 * pio2_2: second 33 bit of pi/2
33 * pio2_2t: pi/2 - (pio2_1+pio2_2)
34 * pio2_3: third 33 bit of pi/2
35 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
36 */
37static const double
38toint = 1.5/EPS,
39pio4 = 0x1.921fb54442d18p-1,
40invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
41pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
42pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
43pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
44pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
45pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
46pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
47
48/* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
49int __rem_pio2(double x, double *y)
50{
51 union {double f; uint64_t i;} u = {x};
52 double_t z,w,t,r,fn;
53 double tx[3],ty[2];
54 uint32_t ix;
55 int sign, n, ex, ey, i;
56
57 sign = u.i>>63;
58 ix = u.i>>32 & 0x7fffffff;
59 if (ix <= 0x400f6a7a) { /* |x| ~<= 5pi/4 */
60 if ((ix & 0xfffff) == 0x921fb) /* |x| ~= pi/2 or 2pi/2 */
61 goto medium; /* cancellation -- use medium case */
62 if (ix <= 0x4002d97c) { /* |x| ~<= 3pi/4 */
63 if (!sign) {
64 z = x - pio2_1; /* one round good to 85 bits */
65 y[0] = z - pio2_1t;
66 y[1] = (z-y[0]) - pio2_1t;
67 return 1;
68 } else {
69 z = x + pio2_1;
70 y[0] = z + pio2_1t;
71 y[1] = (z-y[0]) + pio2_1t;
72 return -1;
73 }
74 } else {
75 if (!sign) {
76 z = x - 2*pio2_1;
77 y[0] = z - 2*pio2_1t;
78 y[1] = (z-y[0]) - 2*pio2_1t;
79 return 2;
80 } else {
81 z = x + 2*pio2_1;
82 y[0] = z + 2*pio2_1t;
83 y[1] = (z-y[0]) + 2*pio2_1t;
84 return -2;
85 }
86 }
87 }
88 if (ix <= 0x401c463b) { /* |x| ~<= 9pi/4 */
89 if (ix <= 0x4015fdbc) { /* |x| ~<= 7pi/4 */
90 if (ix == 0x4012d97c) /* |x| ~= 3pi/2 */
91 goto medium;
92 if (!sign) {
93 z = x - 3*pio2_1;
94 y[0] = z - 3*pio2_1t;
95 y[1] = (z-y[0]) - 3*pio2_1t;
96 return 3;
97 } else {
98 z = x + 3*pio2_1;
99 y[0] = z + 3*pio2_1t;
100 y[1] = (z-y[0]) + 3*pio2_1t;
101 return -3;
102 }
103 } else {
104 if (ix == 0x401921fb) /* |x| ~= 4pi/2 */
105 goto medium;
106 if (!sign) {
107 z = x - 4*pio2_1;
108 y[0] = z - 4*pio2_1t;
109 y[1] = (z-y[0]) - 4*pio2_1t;
110 return 4;
111 } else {
112 z = x + 4*pio2_1;
113 y[0] = z + 4*pio2_1t;
114 y[1] = (z-y[0]) + 4*pio2_1t;
115 return -4;
116 }
117 }
118 }
119 if (ix < 0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */
120medium:
121 /* rint(x/(pi/2)) */
122 fn = (double_t)x*invpio2 + toint - toint;
123 n = (int32_t)fn;
124 r = x - fn*pio2_1;
125 w = fn*pio2_1t; /* 1st round, good to 85 bits */
126 /* Matters with directed rounding. */
127 if (predict_false(r - w < -pio4)) {
128 n--;
129 fn--;
130 r = x - fn*pio2_1;
131 w = fn*pio2_1t;
132 } else if (predict_false(r - w > pio4)) {
133 n++;
134 fn++;
135 r = x - fn*pio2_1;
136 w = fn*pio2_1t;
137 }
138 y[0] = r - w;
139 u.f = y[0];
140 ey = u.i>>52 & 0x7ff;
141 ex = ix>>20;
142 if (ex - ey > 16) { /* 2nd round, good to 118 bits */
143 t = r;
144 w = fn*pio2_2;
145 r = t - w;
146 w = fn*pio2_2t - ((t-r)-w);
147 y[0] = r - w;
148 u.f = y[0];
149 ey = u.i>>52 & 0x7ff;
150 if (ex - ey > 49) { /* 3rd round, good to 151 bits, covers all cases */
151 t = r;
152 w = fn*pio2_3;
153 r = t - w;
154 w = fn*pio2_3t - ((t-r)-w);
155 y[0] = r - w;
156 }
157 }
158 y[1] = (r - y[0]) - w;
159 return n;
160 }
161 /*
162 * all other (large) arguments
163 */
164 if (ix >= 0x7ff00000) { /* x is inf or NaN */
165 y[0] = y[1] = x - x;
166 return 0;
167 }
168 /* set z = scalbn(|x|,-ilogb(x)+23) */
169 u.f = x;
170 u.i &= (uint64_t)-1>>12;
171 u.i |= (uint64_t)(0x3ff + 23)<<52;
172 z = u.f;
173 for (i=0; i < 2; i++) {
174 tx[i] = (double)(int32_t)z;
175 z = (z-tx[i])*0x1p24;
176 }
177 tx[i] = z;
178 /* skip zero terms, first term is non-zero */
179 while (tx[i] == 0.0)
180 i--;
181 n = __rem_pio2_large(tx,ty,(int)(ix>>20)-(0x3ff+23),i+1,1);
182 if (sign) {
183 y[0] = -ty[0];
184 y[1] = -ty[1];
185 return -n;
186 }
187 y[0] = ty[0];
188 y[1] = ty[1];
189 return n;
190}